Short answer: No. The eleven Finger Lakes in west-central New York are separate glacial troughs, not a connected underground network.
During the last Ice Age (~10,000 years ago), massive southward-flowing glaciers carved deep, parallel U-shaped valleys. When the ice retreated, these valleys filled with meltwater, forming long, narrow, deep lakes oriented north–south. Each lake sits in its own bedrock basin; there are no natural subterranean channels linking them.
The New York State Canal System (Erie Canal, Cayuga–Seneca Canal) connects some lakes at the surface for navigation, but these are artificial, shallow channels—not underwater tunnels.
The lakes' striking parallel alignment and great depths fuel speculation. But their isolation is precisely why each has distinct ecology, temperature profiles, and even wine regions (the Finger Lakes AVA).